The concept of Hilbert space is at the heart of the body of quantum mechanics. In this chapter, we introduce the concept of Hilbert space and develop the algebra and calculus of operators in it. It is preceded by introduction of the notions of ‘linear vector space’ and ‘scalar product space’ on which the concept of Hilbert space is based.

Vector Space

A set V of elements {|u, |vi, |w,…g, called vectors, is said to constitute a vector space V over the field F if it is closed under the operations of addition, and multiplication by scalars in F governed by following axioms:

The symbol ji, introduced by Dirac to denote a vector, is called a ket. In the study of the representation of the vector space, we will see that a vector can be represented in different ways. A representation of interest is the one in which it is represented by a column vector. We denote column vectors by a tilde under a letter. Thus, for example,ṵ ewould represent a column vector.

Because of its properties being similar to those of the scalar zero, the null vector j0i may be denoted by the scalar 0. Whether 0 in an expression involving vectors refers to the null vector j0i or to the scalar 0 is to be inferred from the context.

The vector spaces of interest to us are:

1. If the field F in the axioms defining the vector space V is the field of real numbers, then it is called the vector space over the field of real numbers or simply a real vector space.

Entanglement characterizes such correlations between the subsystems of a system which possess exclusive quantum features, in the sense to be described in this chapter. The concept of exclusive quantum features, often called quantumness, stems from the local hidden variables (LHV) theory. We describe in particular the approaches to identify the quantumness of correlations between two spin-1/2 particles. Of various such approaches, the one based on the concept of joint quasiprobability (JQP) distribution provides necessary and sufficient condition for the said identification. The quantumness of correlations between spin-1/2 particles is a resource in quantum computation and quantum information.

Entanglement

We discuss the concept of only the bipartite entanglement, i.e., entanglement between two subsystems of a system. Let the subsystems be named A and B. Let the state of the combined system A + B be described by the density operator. Let and be the operators acting only on A and only on B, respectively. If

for all and then A and B are said to be uncorrelated. The equation above will hold if

where is the density matrix of A alone and that for B alone. For a pure state of the combined system, the condition (18.2) for (18.1) to hold is equivalent with the condition

This chapter discusses the problem of solving the Schrödinger equation for the potentials which are central: a potential V(r) is called central if it depends on the magnitude jrj of r and not on its orientation. We discuss, in particular, the problems of a free particle, potential well, hydrogen atom and of an isotropic harmonic oscillator.

In the case of central potentials, it is convenient and instructive to work in the spherical polar coordinate system discussed next.

Spherical Polar Coordinates

Let (x, y, z) be the coordinates of a point P with respect to the Cartesian system centred at O with ex, ey, ez as the unit vectors along the three axes.

The spherical polar system is described in terms of the coordinates (r, h, ɸ) where, as shown in Figure 14.1, r = jrj is the magnitude of the position vector of P, θ is the angle between OP and the z-axis, and _ is the angle between the x-axis and the projection of OP on the x − y plane. The spherical polar coordinates of P are related to its Cartesian coordinates by the relations

In spherical coordinate description, a potential is central if it is a function only of r. We list below some results useful for the discussion to follow.

A problem of considerable interest is the theory of motion of electrons in a bulk or in a nano sized crystalline solid. Whereas the potential in a bulk solid may be considered to be periodic in entire space, that in a nano sized solid is periodic in a finite region of space. In this chapter, we develop the formalism to study the motion of a particle in a one-dimensional potential which is periodic in a finite part of the space, and the one which is periodic in entire space. In the limit of the potential extending from −1to1, the results for the potentials periodic in the finite part of space reproduce those for the potentials periodic in entire space. We will see that completely periodic potentials can also be treated invoking symmetry considerations.

Potential Periodic in a Finite Region of Space

Consider a potential V(x) which is periodic with period a in the finite region of space 0 ≤ x ≤ Na such that

whereas on the remaining part of the x-axis,

The solution of the Schrödinger equation corresponding to the potential defined in (12.1) and (12.2) is obtained by solving it in different regions and matching the solutions at the boundaries of the adjoining regions.

We find first the solution in the region 0 ≤ x ≤ Na in which the potential is periodic. Consider the interval na ≤ x ≤ (n + 1)a which we call the nth interval. Since, due to (12.1), the potential in nth interval is same as that in (0, a), the linearly independent solutions of the Schrödinger equation in nth interval are same as those in (0, a).

History is the most fundamental science, for there is no human knowledge which cannot lose its scientific character when men forget the conditions under which it originated, the questions it answered, and the functions it was created to serve.Astudent of quantum mechanics can ignore these words of Benjamin Farrington, often quoted by Schrödinger [1], only at his own peril. For, without their historical perspective, the counter-intuitive nature of the physical laws of the quantum theory may appear to be mystic. Indeed, the laws of quantum mechanics exhibit their distinctive character on a scale which is out of reach of our everyday experience. This is in contrast with the intuitive appeal of the Newton's laws and their conformity with our common experience. However, the inability of the then known theories, referred to now as the classical theories, to explain certain physical phenomena led to the search for a new theory. It culminated with the advent of the quantum theory. The true scientific character of the quantum theory cannot be appreciated without understanding the conditions under which it originated, the questions it answered, and the functions it was created to serve. Such an understanding can even motivate one to ponder whether there can be any other description of the laws of nature which is free from the conceptual and philosophical inconsistencies which are often pointed out in the present-day quantum theory. This chapter outlines brie_y how the ideas leading to the birth of quantum mechanics took shape.

This book is the outcome of my belief, shared undoubtedly by many that, like classical mechanics is taught starting with Newton's laws, so should the teaching of quantum mechanics be based on its postulates, and that since, in contrast with the intuitive appeal of Newton's laws, the quantum laws appear counter-intuitive, therefore, introduction of the postulates of quantum mechanics must be preceded by narrating the fascinating story of their birth. The understanding of the thought process that led to seemingly puzzling but remarkably accurate formulation of the laws of nature can arguably provide greater appreciation and grasp of the essence of radically different and counter-intuitive ideas of the theory. Furthermore, I find that some mathematical techniques, which can significantly simplify the algebra, are generally overlooked in the standard quantum mechanics literature.

The subject is generally introduced by glancing through those phenomena which historically necessitated search for a new theory and by providing the motivation and justification of the new ideas by examples. Making a departure from that approach, drawing upon the papers that formed the foundations of quantum mechanics, this book opens by providing an account of how the new theory took shape. It describes, for example, how Planck came up with the law of black-body radiation. How Heisenberg came up with the idea of representing a dynamical variable x by the set of quantities [xmnm] labelled by two indices and how he arrived at the law of their multiplication, not knowing that his was the matrix representation of the classical dynamical variables.

The non-linear equat

where the coefficients p(t), b(t), q(t) are functions of the real variable t, is known as the Riccati equation. The solution of (C.1) is known only for some special forms of the coefficients in it.

By transforming to the variable x(t) defined by

the Riccati equation transforms to the second-order ordinary linear differential equation for x(t):

The solution of the Riccati equation may thus be used to solve (C.3).

A solvable case of (C.1) arises when the coefficients in it are independent of t. Assuming that to be the case, rewrite (C.1) as

Substitute (C.5) in (C.4) and integrate the resulting equation to obtain

On rearranging the terms, the expression for y(t) reads

The same result can be arrived at by solving the linear equation (C.3) corresponding to (C.1) (see Ex. C.2)

Ex. C.1. Show that even though x(t) in (C.2) is the linear combination of two linearly independent solutions x1(t) and x2(t) of (C.3), y(t) is determined only by its initial value.

Ex. C.2. Assuming the coefficients in (C.1) to be independent of t, solve the associated linear equation (C.3) for x(t) and show that the solution y(t) so obtained is the same as (C.8).

This chapter summarizes some qualitative properties of the solutions of the one-dimensional Schrödinger equation. The one-dimensional equation represents not only the idealized situations but is of interest also for solving the Schrödinger equation for three-dimensional central potentials as in that case the problem reduces to solving one-dimensional equations. The study of the qualitative properties enables one to extract useful physics information without solving the equation which more often than not is a formidable task. The question of identification of exactly solvable one-dimensional potentials and their explicit solutions is addressed in subsequent chapters.

Asymptotic Behaviour

Consider a particle of mass m constrained to move in one dimension in a time-independent potential V(x). Its wave function of definite energy E solves the Schrödinger equation

where U(x) and are the scaled potential and energy defined by

We study first the asymptotic properties of the solutions of (9.1) under the condition as , where V± are constants. The solution of (9.1) as is then given by

where. It is real for. In that case, as we will see in Section 10.1, the first term in (9.3) represents the particle moving freely in the direction, whereas the second one represents the one moving freely in the −x direction.

However, if E < V+m, then k+ is imaginary and (9.3) reads

As a function of the variable x the Dirac delta function, or simply the delta function, denoted by δ(x), is defined as follows:

The condition on the ei's means that the limit of integration in (A.1) should include origin. The ϵ1, ϵ2 in (A.1) can have any positive value and may even be infinitesimally close to zero. It then follows that for f (x) continuous at the origin (a, b, ϵ > 0, ϵ ½ 0),

The second equation above follows by the use of the first part of (A.1), and the third from the fact that, due to continuity of f (x) at x = 0, the value of f (x) for x 2 (−ϵ, ϵ) can be taken to be f (0) as ϵ → 0. In general

The derivative of the delta function is defined as follows:

The higher derivatives can be defined in similar manner.

The defining relations of the delta function and its derivative may be written symbolically as follows:

where prime denotes the derivative with respect to x.

We may similarly define the three-dimensional delta function δ(3)(r−r), where r, rÌ are the position vectors, by the equation

where the integral is over the volume containing r. Let us express δ(3)(rÌ − r) in terms of one-dimensional delta functions. We will see that the said expression depends on the coordinate system.

Whether classical or quantum, solving the equations of motion analytically exactly is often a formidable task. Some properties of the motion may, however, be inferred without solving the equations of motion. One such important property is the symmetry of the Hamiltonian under some transformation. The interesting and practically useful aspect of symmetry is that it characterizes the constants of the motion. The symmetry may be under some transformation of space and time, called the geometrical symmetry, or it may be in some abstract space resulting from particular functional form of the Hamiltonian, called the dynamical symmetry. This chapter investigates the consequences of such symmetries on the quantum mechanical description of the motion of a particle.

Symmetry Transformation

Consider a system whose state for some observer is. If the observer performs the measurement of an observable on the system, we know that the probability of observing an eigenvalue corresponding to an eigenstate as an outcome is. Now, let the same experiment be described in a coordinate system transformed with respect to the first one. Since the result of a measurement is not expected to depend on the coordinate system used, the probability of observing certain eigenvalue as the outcome should not change. Generalizing the arguments above, we expect the transition probability between the states to be independent of the coordinate system. This chapter is concerned with investigating the consequences of the invariance of the probabilities under linear transformation of the coordinate system.

The algebra of operators in the space of functions is of considerable importance in quantum mechanics. The function space of interest in quantum mechanics is the one in which the functions are square integrable in the sense described below. In this chapter, we study the algebra of operators in the space of square integrable functions.

Space of Square Integrable Functions

Consider the space of complex-valued functions of a real variable x (a ≤ x ≤ b) in which the scalar product between the functions g(x) and f (x), denoted by (g(x), f (x)), is defined by

As a consequence of the definition of scalar product given above,

is finite. A function f (x) for which the integral in the equation above is finite is said to be square integrable. It may be verified that the definition of the scalar product given above satisfies all the axioms of scalar product, namely,

In particular, as a consequence of the axioms above, follows the Schwarz inequality (see (2.8)) which, in the present case, assumes the form

An important consequence of this inequality is the result derived in Ex. 4.1 establishing that any linear combination of square integrable functions is also square integrable, which asserts that the space of square integrable functions forms a vector space.

This chapter identifies the class of continuously varying one-dimensional potentials for which the time-independent Schrödinger equation can be solved analytically exactly by reducing it to the hypergeometric or the confluent hypergeometric form by means of the transformation of the dependent and the independent variables. Most of the widely studied exactly solvable problems in quantum mechanics are special cases of the potentials in the said classes. The problem of finding analytically exact solution of the Schrödinger equation by reducing it to the hypergeometric or the confluent hypergeometric form has been addressed in different ways, for example, in [58–61].

To that end, we start with the one-dimensional Schrödinger equation (9.1) for the wave function of a particle of mass m and energy E in the potential V(x),

where the ‘prime’ denotes differentiation with respect to the independent variable.We consider first the case when the functions f (z) and u(z) are independent of “ and for which (11.4) assumes the form of either the (i) hypergeometric or (ii) the con_uent hypergeometric equation.We will subsequently consider the class of potentials for which (11.4) is reducible to the con_uent hypergeometric form under energy-dependent transformation.

Potentials for which Schrödinger Equation is Reducible to Hypergeometric Form

By assuming f (z) and u(z) to be independent of, in this section we find the form of the potential U(z) for which (11.4) assumes the form of the hypergeometric equation.

As discussed in the Appendix B, a second-order ordinary differential equation is reducible to the hypergeometric equation if it has three regular singularities, including the singularity, if any, at infinity and no other singularity.

Some Mathematical Formulas

This chapter discusses the quantum mechanical formulation in the position and the momentum representations. The position representation provides a natural means of constructing the quantum analog of a classical mechanical problem. The momentum representation emerges as the Fourier transform of the position representation and often provides a convenient means of studying the time evolution of the states.

Position Representation

For the sake of simplicity we consider first a particle constrained to move on the real axis. Recall that, according to the second postulate of quantum mechanics, to every classical dynamical variable there corresponds a unique hermitian operator. Accordingly, let the hermitian operator corresponding to classical position x be denoted by. Let be an eigenstate of the position operator corresponding to the eigenvalue x, i.e.,

where the second equation above is due to the hermiticity of. A state

in the state space of the particle is then represented by the function

called the wave function. By virtue of the axioms of the scalar product,. The representation of states and operators in the basis of the eigenstates fjxig of O x is called the position representation. As is expected of a self-adjoint operator, the eigenstates fjxig of O x constitute complete orthonormal set. The completeness relation reads

The orthonormality relation reads Hence, in terms of its position representative x(x), any state can be represented as

Now, we know from the fourth postulate that is the probability density which, on invoking (8.2), implies that

According to the first postulate of quantum mechanics, the state of an isolated system is described by a vector in the Hilbert space. However, more often than not, we encounter systems interacting with other systems. In this chapter we address the question of describing the state of a system interacting with other systems without any reference to the details of the other system or that of the interaction between them. This leads to the concept of the density operator. The concept of density operator will be seen to arise also while characterizing the state of a mixture of particles in different states.

State of a Subsystem

Consider a system composed of the subsystems A and B lying in the Hilbert space of dimensions nA and nB, respectively. We use the formalism developed in Section 5.6 for describing a composite system. Let the state of the composite system be. Recall (5.71) which expresses the composite state of A and B in terms of the superposition of the tensor product of their basis states:

where and are the sets of the orthonormal basis states of A and B. Let it be that we are interested in the properties of only one of the subsystems, say, the subsystem A. Those properties are determined by the expectation values of the operators which act on the states of the subsystem A alone.